## The Scaled Beta2 Distribution as a robust prior for scales, and a Explicit Horseshoe Prior for locations

##### Abstract

In the tradition of ”Conjugate Priors” analysis (and extensions to Conditional Conjugate Priors), priors for locations are assumed Normal and for precisions are assumed Gamma or equivalently for variances Inverted Gamma. Following an idea of deFinnetti (1961), by regarding each observation (or set of observations) as having its own precision taken from a super-population of scales distributed as a Gamma, the Student distribution, as a scale-mixture of Normals, replaces the Normal Gaussian Model. This procedure adds various forms of robustness to priors and models. In a parallel manner,
it is natural to think of the sum of squares as having a conditional Gamma distribution, and mixing scales again by a gamma distribution we obtain what we call the Scaled
Beta2 Distribution (SBeta2). We propose here this distribution as an alternative to the ubiquitous Inverted-Gamma IG prior for variances and standard deviations, or
more generally for scales. Vague IG priors as a quasi-non informative prior for reciprocal of scales with very low hyper-parameters, say I G(0.001, 0.001) percolates most
of Bayesian Statistics. It is known however, as pointed out by Gelman (2006) (but see Berger (2006) to mathematically explain the phenomenon) that far from being quasi non-informative the ”vague” IG, leads to very low variances of the eﬀects and very strong shrinkages to the general mean. Gelman proposes Uniform and Half-Cauchy priors. In the same vein, we propose here the Scaled Beta2 family. Advantages of the SBeta2 are numerous, among them are: its ﬂexibility, and for particular values of the hyper-parameters it can be as heavy or heavier tailed as the half Cauchy, and also different behavior at the origin can be modeled. Furthermore it is easy to simulate from, and it can be inside a Gibbs sampling scheme. Besides, both the scale or the square of the scale, (or their reciprocals) cam be modeled as a SBeta2, both for Normals or other Likelihoods like in the Student-t family. There is an additional bonus: when the conditional prior for the location is Cauchy for example, and the prior for the scale,
is SBeta2,the marginal for the location can be obtained proportional of well known functions, and for integer values of the hyper-parameters of the SBeta2, the marginal densities for locations and cumulative distributions can be found explicitly. This adds insight in the analysis and make it easier to elicit the prior scale, which is very convenient in general and for Empirical Bayes modeling. When the scale is modeled as a SBeta2, the marginal prior location has the shape of a horseshoe prior, Carvalho et. al (2010) and to the best of our knowledge, it is the ﬁrst horseshoe prior that can be
shown in closed form. We call these priors, the explicit Horseshoe priors. Horseshoe priors are also speciﬁcally well suited for full Bayes Hierarchical modeling, leading to
strong shrinkage when there are is no conﬂict between data and priors, and discounting the prior when there is conﬂict. Alternatively, if instead of the scale, the square of the scale is assumed as a SBeta2, then Fúquene, Pérez and Pericchi (2011) obtained what is called a Student-Scaled Beta2 marginal prior, known in closed form, which is also
very convenient as a heavy tailed prior for the location.